cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A356709 Numbers k such that Mordell's equation y^2 = x^3 + k^3 has exactly 1 integral solution.

Original entry on oeis.org

3, 5, 6, 12, 13, 15, 17, 19, 20, 24, 27, 29, 30, 31, 39, 41, 42, 43, 45, 47, 48, 51, 52, 53, 54, 55, 58, 59, 60, 61, 62, 66, 67, 68, 69, 73, 75, 76, 77, 79, 80, 82, 83, 85, 87, 89, 93, 94, 96, 97, 101, 102, 103, 106, 107, 108, 109, 111, 113, 115, 116, 117, 118, 119
Offset: 1

Views

Author

Jianing Song, Aug 23 2022

Keywords

Comments

Numbers k such that Mordell's equation y^2 = x^3 + k^3 has no solution other than the trivial solution (-k,0).
Cube root of A179145.

Examples

			3 is a term since the equation y^2 = x^3 + 3^3 has no solution other than (-3,0).

Links

Crossrefs

Cf. A081119, A179145, A179147, A179149, A179151, A356710, A356711, A356712.
Indices of 1 in A356706, of 0 in A356707, and of 1 in A356708.
Complement of A356720.
Cf. also A356713, A228948.

A356706 Number of integral solutions to Mordell's equation y^2 = x^3 + n^3.

Original entry on oeis.org

5, 7, 1, 5, 1, 1, 3, 9, 5, 5, 3, 1, 1, 5, 1, 5, 1, 7, 1, 1, 3, 3, 3, 1, 5, 3, 1, 5, 1, 1, 1, 9, 5, 3, 3, 5, 5, 3, 1, 5, 1, 1, 1, 3, 1, 3, 1, 1, 5, 7, 1, 1, 1, 1, 1, 7, 7, 1, 1, 1, 1, 1, 3, 5, 17, 1, 1, 1, 1, 5, 3, 9, 1, 3, 1, 1, 1, 9, 1, 1, 5, 1, 1, 5, 1, 3, 1, 5, 1, 5, 5, 3, 1, 1, 3, 1, 1
Offset: 1

Views

Author

Jianing Song, Aug 23 2022

Keywords

Examples

			a(8) = 9 since the equation y^2 = x^3 + 8^3 has 9 integral solutions (-8,0), (-7,+-13), (4,+-24), (8,+-32), and (184,+-2496).

Links

Crossrefs

Cf. A081119, A356707, A356708.
Indices of 1, 3, 5, and 7: A356709, A356710, A356711, A356712.

Programs

  • SageMath
    [len(EllipticCurve(QQ, [0, n^3]).integral_points(both_signs=True)) for n in range(1, 61)] # Lucas A. Brown, Sep 03 2022

Formula

a(n) = A081119(n^3).

Extensions

a(21) corrected and a(22)-a(60) from Lucas A. Brown, Sep 03 2022
Terms a(61) onward from Max Alekseyev, Jun 01 2023

A356711 Numbers k such that Mordell's equation y^2 = x^3 + k^3 has exactly 5 integral solutions.

Original entry on oeis.org

1, 4, 9, 10, 14, 16, 25, 28, 33, 36, 37, 40, 49, 64, 70, 81, 84, 88, 90, 91, 100, 104, 121, 126, 130, 132, 140, 144, 154, 160, 169, 176, 184, 193, 196
Offset: 1

Views

Author

Jianing Song, Aug 23 2022

Keywords

Comments

Cube root of A179149.
Contains all squares: suppose that y^2 = x^3 + t^6, then (y/t^3)^2 = (x/t^2)^3 + 1. The elliptic curve Y^2 = X^3 + 1 has rank 0 and the only rational points on it are (-1,0), (0,+-1), and (2,+-3), so y^2 = x^3 + t^6 has 5 solutions (-t^2,0), (0,+-t^3), and (2*t^2,+-3*t^3).

Examples

			1 is a term since the equation y^2 = x^3 + 1^3 has 5 solutions (-1,0), (0,+-1), and (2,+-3).

Crossrefs

Cf. A081119, A179145, A179147, A179149, A179151, A356709, A356710, A356712.
Indices of 5 in A356706, of 2 in A356707, and of 3 in A356708.

Extensions

a(31)-a(35) from Max Alekseyev, Jun 01 2023

A356708 Number of integral solutions to Mordell's equation y^2 = x^3 + n^3 with y nonnegative.

Original entry on oeis.org

3, 4, 1, 3, 1, 1, 2, 5, 3, 3, 2, 1, 1, 3, 1, 3, 1, 4, 1, 1, 2, 2, 2, 1, 3, 2, 1, 3, 1, 1, 1, 5, 3, 2, 2, 3, 3, 2, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 3, 4, 1, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 1, 2, 3, 9, 1, 1, 1, 1, 3, 2, 5, 1, 2, 1, 1, 1, 5, 1, 1, 3, 1, 1, 3, 1, 2, 1, 3, 1, 3, 3, 2, 1, 1, 2, 1, 1, 4, 2, 3
Offset: 1

Views

Author

Jianing Song, Aug 23 2022

Keywords

Comments

Equivalently, number of different values of x in the integral solutions to the Mordell's equation y^2 = x^3 + n^3.

Examples

			a(2) = 4 because the solutions to y^2 = x^3 + 2^3 with y >= 0 are (-2,0), (1,3), (2,4), and (46,312).

Crossrefs

Cf. A081119, A134108, A356706, A356707.
Indices of 1, 2, 3, and 4: A356709, A356710, A356711, A356712.

Programs

  • SageMath
    [(len(EllipticCurve(QQ, [0, n^3]).integral_points(both_signs=True))+1)/2 for n in range(1, 61)] # Lucas A. Brown, Sep 04 2022

Formula

a(n) = (A081119(n^3)+1)/2 = A134108(n^3) = (A356706(n)+1)/2 = A356707(n)+1.

Extensions

a(21) corrected and a(22)-a(60) by Lucas A. Brown, Sep 04 2022
a(61)-a(100) from Max Alekseyev, Jun 01 2023

A356710 Numbers k such that Mordell's equation y^2 = x^3 + k^3 has exactly 3 integral solutions.

Original entry on oeis.org

7, 11, 21, 22, 23, 26, 34, 35, 38, 44, 46, 63, 71, 74, 86, 92, 95, 99, 110, 122, 129, 136, 152, 155, 158, 170, 175, 177, 183, 189, 190, 198, 201, 203, 207, 211
Offset: 1

Views

Author

Jianing Song, Aug 23 2022

Keywords

Comments

Cube root of A179147.

Examples

			7 is a term since the equation y^2 = x^3 + 7^3 has 3 solutions (-7,0) and (21,+-98).

Crossrefs

Cf. A081119, A179145, A179147, A179149, A179151, A356709, A356711, A356712.
Indices of 3 in A356706, of 1 in A356707, and of 2 in A356708.

Extensions

a(30)-a(36) from Max Alekseyev, Jun 01 2023

A356712 Numbers k such that Mordell's equation y^2 = x^3 + k^3 has exactly 7 integral solutions.

Original entry on oeis.org

2, 18, 50, 56, 57, 98, 112, 114, 148, 162, 224, 228, 273, 280, 330, 336, 338, 364, 448, 504, 513, 578
Offset: 1

Views

Author

Jianing Song, Aug 23 2022

Keywords

Comments

Cube root of A179151.

Examples

			2 is a term since the equation y^2 = x^3 + 2^3 has 3 solutions (-2,0), (1,+-3), (2,+-4), and (46,+-312).

Crossrefs

Cf. A081119, A179145, A179147, A179149, A179151, A356709, A356710, A356711.
Indices of 7 in A356706, of 3 in A356707, and of 4 in A356708.

A356704 a(n) is the least k such that Mordell's equation y^2 = x^3 + k^3 has exactly 2*n+1 integral solutions.

Original entry on oeis.org

3, 7, 1, 2, 8, 329, 217, 506, 65, 260, 585
Offset: 0

Views

Author

Jianing Song, Aug 23 2022

Keywords

Comments

a(n) is the least k such that y^2 = x^3 + k^3 has exactly n solutions with y positive, or exactly n+1 solutions with y nonnegative.
a(n) is the smallest index of 2*n+1 in A356706, of n in A356707, and of n+1 in A356708.

Examples

			a(4) = 8 since y^2 = x^3 + 8^3 has exactly 9 solutions (-8,0), (-7,+-13), (4,+-24), (8,+-32), and (184,+-2496), and the number of solutions to y^2 = x^3 + k^3 is not 9 for 0 < k < 8.

Crossrefs

Cf. A081119, A081120, A179162, A179175, A356705, A356706, A356707, A356708.

Formula

a(n) = A179162(2*n+1)^(1/3).
Showing 1-7 of 7 results.

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